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Intermediate Algebra Examples
Basic Formulas

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1. Find the area of a triangle [ ] with base 12 and height 7.
 Use the formula with b = 12 and h = 7 to obtain triangle area calculation
The area of the triangle is 42 sq. units.

2. What is the area of the trapezoid [ trapezoid interactive trapezoid area interactive ]shown in Fig.1? trapezoid
 Use the formula Formula for area of a trapezoid to obtain area calculations
The area of the trapezoid is 13/2 sq. units.

annulus 3. What is the area of the blue annulus [ annulus general information annulus area ] shown in Fig. 2?
The area of the blue annulus can be computed by subtracting the area of the smaller circle from the area of the larger circle. Use the formula formula for the area of circle to do the calculations.
annulus computations
The area of the annulus is 224 pi sq. units.


 

 

land parcel 4. Find the area of the parcel of land pictured and dimensioned in Fig. 4. Express the area in acres correct to hundredths of an acre.
This parcel of land is in the shape of a quadrilateral but not a quadralateral for which we have a formula for the area. Therefore we will use the strategy given by the wise old owl. We can envision the parcel inscribed in a rectangle (red rectangle in Fig. 5) which contains two triangles (green triangles in Fig. 5) in addition to the parcel of land.

land parcel inside rectangle
An examination of Fig. 5 shows the desired area of the parcel of land (the white quadralateral in Fig. 5) is the area of the red rectangle minus the areas of the two green triangles. Because we know formulas for the areas of rectangles and triangles, this seems like a good strategy.

However, before proceeding to the computations we must determine the missing dimensions of the triangles and the rectangle. The formulas we plan to use assume that all dimensions use the same units of measure. Therefore we must convert miles to feet.
These modifications are shown in blue on Fig. 6.
land parcel inside rectangle
Use AP to represent the desired area of the parcel of land.
Use AR to represent the area of red rectangle.
Use AT1 to represent the area of green triangle 1.
Use AT2 to represent the area of green triangle 2.
Then AP = AR - (AT1 + AT2) will provide the desired area.
Recall the formula for the area of a rectangle (area of a rectangle)
and the area of a triangle (area of a triangle).
A bit of deductive reasoning (from general formula to specific areas) yields:
AR = (15840)(300) = 4752000 sq.ft.
AT1 = (0.5)(15840)(100) = 792000 sq.ft.
AT2 = (0.5)(1056)(300) = 158400 sq.ft.
AP = AR - (AT1 + AT2) = 4,752,000 - (792,000 + 158400) = 3,801,600 sq.ft.

Finally we must convert 3,801,600 sq.ft. to acres.
Divide 3,801,600 by 43,560 to obtain 87.272727 acres. When this number is rounded to hundredths we obtain 87.27 acres as the area of the parcel of land.

 

 

5. A goat is tied to the corner of a 12-by-15 foot building with a rope 10 ft. long. What is the area the goat may graze?

goat grazing Sketch a diagram which models the problem. The picture need not be to scale, but the dimensions must be correct and it must accurately model the problem. Fig. 7 is such a picture. Observe that Fig. 7 may contain more detail than is required to analyze this problem.
It is presumed that the angles at the corners A, B, C, and D of the shed are each 90°. From the assumption that angle A is 90°, it follows that angle θ is 270° which is three quarters of a circle.
We have just solved the problem! The area that the goat may graze is three-quarters of the area of a 10 ft. circle. Now all we need is a few computations.
Let A be the area grazed by the goat. Then

goat grazing computations
This is the exact area that the goat may graze.
To the nearest sq.ft. the area is 236 sq.ft.

 

6. A goat is tied to the corner of a 12-by-15 foot building with a rope 20 ft. long. What is the area the goat may graze?

goat grazing In view of Example 5, we might be tempted to believe the area is three-quarters of the area of a 20 ft. circle. However, the sketch shown in Fig. 8 demonstrates that a simple change in the length of the tether significantly changes the problem.
The desired area still involves three-quarters of the area of a 20 ft. circle, but it also includes one-quarter of the area of a 5 ft. circle as well as one-quarter of the area of an 8 ft. circle.
Now all we need is a few computations.
Let A be the total area the goat may graze.
Let AM be three-quarters of the area of a 20 ft. circle (labeled M in Fig. 8)
Let AN be one-quarter of the area of a 5 ft. circle (labeled N in Fig. 8)
Let AP be one-quarter of the area of a 8 ft. circle (labeled P in Fig. 8)
Then A = AM + AN + AP is a formula for the desired area.
The computations should be constructed and presented as indicated by this formula.

goat grazing 2
The goat may graze exactly goat grazing 2 exact area

 

 

Examples 5 and 6 were inspired by an example on the excellent Purplemath.com website.
Stapel, Elizabeth. "Geometry Word Problems: The Box Problem & The Goat Problem."
Purplemath. Available from http://www.purplemath.com/modules/perimetr5.htm. Accessed 08 January 2010.